(Eternal) Vertex Cover Number of Infinite and Finite Grid Graphs

TL;DR

This paper investigates the eternal vertex cover problem on regular grid graphs, providing bounds on the minimum number of guards needed for both infinite and finite grids, and generalizing concepts for infinite cases.

Contribution

It introduces generalized notions of vertex cover and eternal vertex cover for infinite grids and establishes bounds for finite and infinite grid graphs.

Findings

Derived tight bounds for guards on finite grids

Established bounds for guards on infinite grids

Unified concepts for infinite and finite grid covers

Abstract

In the eternal vertex cover problem, mobile guards on the vertices of a graph are used to defend it against an infinite sequence of attacks on its edges by moving to neighbor vertices. The eternal vertex cover problem consists in determining the minimum number of necessary guards. Motivated by previous literature, in this paper, we study the vertex cover and eternal vertex cover problems on regular grids, when passing from infinite to finite version of the same graphs, and we provide either coinciding or very tight lower and upper bounds on the number of necessary guards. To this aim, we generalize the notions of minimum vertex covers and minimum eternal vertex cover in order to be well defined for infinite grids.